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This paper investigates the robust stability of an asymptotic second-order sliding mode (2nd-SM) control system, where a first-order sliding mode (1st-SM) control law is implemented to realize an asymptotic 2nd-SM control for a linear time-invariant continuous-time system with a relative degree of two. It is found in the paper that a 2nd-SM can be reached locally and asymptotically by a 1st-SM control law if the sum of the system poles is less than the sum of the system zeros. The asymptotic convergence to the 2nd-SM and the robust stability of the asymptotic 2nd-SM control system are for the first time proved with Lyapunov functions, in the presence of matched external disturbances and parameter uncertainties. Finally, the effectiveness of the asymptotic 2nd-SM control algorithm is verified through numerical simulations.